FIXED POINT THEOREMS FOR GENERAL CLASSES OF MAPS ACTING ON TOPOLOGICAL VECTOR SPACES

September 30, 2011 9:54 WSPC/INSTRUCTION FILE 01˙Agaregtao

Asian-European Journal of MathematicsVol. 4, No. 3 (2011) 373–387c© World Scientific Publishing CompanyDOI: 10.1142/S1793557111000307

FIXED POINT THEOREMS FOR GENERAL CLASSES OF MAPS

ACTING ON TOPOLOGICAL VECTOR SPACES

Ravi P. Agarwal

Department of Mathematical Sciences, Florida Institute of Technology

150 West University Boulevard, Melbourne Florida 32901

KFUPM Chair Professor, Mathematics and Statistics Department,

King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia

[email protected]

Donal O’Regan

Department of Mathematics, National University of Ireland,

Galway, Ireland

[email protected]

Mohamed-Aziz Taoudi

Universite Cadi-Ayyad, Laboratoire de Mathematiques

et de Dynamique de Populations, Marrakech, Maroc

[email protected]

Communicated by O. ChristensenReceived August 18, 2010Revised November 9, 2010

We present new fixed point theorems for multivalued Ukc -admissible maps acting on

locally convex topological vector spaces. The considered multivalued maps need not becompact. We merely assume that they are weakly compact and map weakly compactsets into relatively compact sets. Our fixed point results are obtained under Schauder,Leray–Schauder and Furi-Pera type conditions. These results are useful in applicationsand extend earlier works.

Keywords: Fixed point theorem; Ukc map; topological vector space.

AMS Subject Classification: 47H10, 47H04, 47H09, 47H14

1. Introduction

In recent years, several fixed point theorems for compact Ukc -admissible maps were

established in topological vector spaces. We quote for instance the contributions of

Park [19], O’Regan [16–18], Agarwal and O’Regan [2] and Agarwal, O’Regan and

373

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http://dx.doi.org/10.1142/S1793557111000307


374 R. P. Agarwal, D. O’Regan & M. A. Taoudi

Park [1]. In the present paper, we establish several new fixed point theorems for

noncompact Ukc maps and discuss boundary conditions. The maps considered will be

weakly compact and map weakly compact sets into relatively compact sets. These

conditions are useful in applications namely in the study of existence of solution

to differential and integral equations and inclusions with lack of compactness (see

for example [13, 14, 21, 22] and the references therein). Our fixed point theorems

are obtained under Schauder, Leray-Schauder, Sadovskii and Furi-Pera boundary

conditions. The analysis relies on fixed point theorems for compact Ukc maps and

uses properties of the Minkowski functional and retractions on topological vector

spaces. Also, using the notion of measure of weak noncompactness, our results

are stated in more general forms in Banach spaces setting. For the remainder of

this section we present some definitions and some known facts. Let X and Y be

topological spaces. A multivalued map F : X → 2Y is a point to set function if

for each x ∈ X, F (x) is a nonempty subset of Y. For a subset M of X we write

F (M) = ∪x∈MF (x) and F−1(M) = x ∈ X : F (x) ∩ M 6= ∅. The graph of

F is the set Gr(F ) = (x, y) ∈ X × Y : y ∈ F (x). We say that F is upper

semicontinuous (u.s.c. for short) at x ∈ X if for every neighborhood V of F (x)

there exists a neighborhood U of x with F (U) ⊆ V (equivalently, F : X → 2Y is

u.s.c. if for any net xα in X and any closed set B in Y with xα → x0 ∈ X and

F (xα) ∩ B 6= ∅ for all α, we have F (x0) ∩ B 6= ∅). We say that F : X → 2Y is

upper semicontinuous if it is upper semicontinuous at every x ∈ X. The function F

is lower semicontinuous (l.s.c.) if the set F−1(B) is open for any open set B in Y .

If F is l.s.c. and u.s.c., then F is continuous.

If Y is compact, and the images F (x) are closed, then F is upper semicontinuous

if and only if F has a closed graph. In this case, if Y is compact, we have that F

is upper semicontinuous if xn → x, yn → y, and yn ∈ F (xn), together imply that

y ∈ F (x).

A nonempty subset X of a Hausdorff topological vector space E is said to be

admissible (in the sense of Klee [12]) provided that, for every compact subset K of

X and every neighborhood V of the origin 0 of E, there exists a continuous map

h : K → X such that x − h(x) ∈ V for all x ∈ K and h(K) is contained in a finite

dimensional subspace of E. Note that every nonempty convex subset of a locally

convex topological vector space is admissible [15]. Other examples of admissible

maps can be found in [19].

Let X be a nonempty, convex subset of a Hausdorff topological vector space

E and Y a topological space. Recall a polytope P in X is any convex hull of a

nonempty finite subset of X.

Suppose E1 and E2 are Hausdorff topological spaces. Given a class χ of maps,

χ(E1, E2) denotes the set of maps F : E1 → 2E2 (nonempty subsets of E2 ) belonging

to χ, and χc the set of finite compositions of maps in χ. A class U of maps is defined

by the following properties:

(1) U contains the class C of single valued continuous functions,

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Fixed Point Theorems for General Classes of Maps 375

(2) each F ∈ Uc is upper semicontinuous and compact valued, and

(3) for any polytope P, F ∈ Uc(P, P ) has a fixed point, where the intermediate

spaces of composites are suitably chosen for each U .

Definition 1.1. F ∈ Ukc (E1, E2) (i.e. F is Uk

c -admissible) if F is closed and if for

any compact subset K of E1, there is a G ∈ Uc(K;E2) with G(x) ⊆ F (x) for each

x ∈ K.

The following result was proved in [19].

Theorem 1.1. Let E be a Hausdorff topological vector space and X an admissible,

convex subset of E. Then any compact map F ∈ Ukc (X,X) has a fixed point.

Recall also that a nonempty topological space is said to be acyclic if all its

reduced Cech homology groups over the rationals are trivial. LetX and Y be subsets

of Hausdorff topological vector spaces E1 and E2 respectively. Let K(Y ) denote the

family of nonempty closed convex subsets of Y. Now F : X → K(Y ) is acyclic if

F is upper semicontinuous with acyclic values. Given two open neighborhoods U

and V of the origins in E1 and E2 respectively, a (U, V )-approximate continuous

selection [5] of F : X → 2Y is a continuous function s : X → Y satisfying

s(x) ∈ (F [(x+ U) ∩X ] + V ) ∩ Y for every x ∈ X.

We say F : X → K(Y ) is approximable if it is a closed map and if its restriction F |K

to any compact subset K of X admits a (U, V )-approximate continuous selection

for every open neighborhood U and V of the origins in E1 and E2 respectively. A

map is said to be AcAp if it is either acyclic or approximable.

Remark 1.1. Notice that Ukc is closed under compositions [16]. The class Uk

c in-

cludes [1] the Kakutani maps, the acyclic maps, the O’Neill maps, the approximable

maps and the maps admissible with respect to Gorniewicz.

Now let X be a Banach space and let B(X) denote the collection of all nonempty

bounded subsets of X and W(X) the subset of B(X) consisting of all weakly

compact subsets of X. Also, let Br denote the closed ball centered at 0 with

radius r.

Definition 1.2. [4] A function ψ : B(X) → R+ is said to be a measure of weak

noncompactness if it satisfies the following conditions:

(1) The family ker(ψ) = M ∈ B(X) : ψ(M) = 0 is nonempty and ker(ψ) is

contained in the set of relatively weakly compact sets of X.

(2) M1 ⊆M2 ⇒ ψ(M1) ≤ ψ(M2).

(3) ψ(co(M)) = ψ(M), where co(M) is the closed convex hull of M.

(4) ψ(λM1 + (1− λ)M2) ≤ λψ(M1) + (1− λ)ψ(M2) for λ ∈ [0, 1].

(5) If (Mn)n≥1 is a sequence of nonempty weakly closed subsets of X with M1

bounded and M1 ⊇ M2 ⊇ . . . ⊇ Mn ⊇ . . . such that limn→∞ ψ(Mn) = 0,

then M∞ :=⋂∞

n=1Mn is nonempty.

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The family kerψ described in (1) is said to be the kernel of the measure of weak

noncompactness ψ. Note that the intersection set M∞ from (5) belongs to kerψ

since ψ(M∞) ≤ ψ(Mn) for every n and limn→∞ ψ(Mn) = 0. Also, it can be easily

verified that the measure ψ satisfies

ψ(Mw) = ψ(M) (1.1)

where Mw is the weak closure of M.

A measure of weak noncompactness ψ is said to be regular if

ψ(M) = 0 if and only if M is relatively weakly compact. (1.2)

subadditive if

ψ(M1 +M2) ≤ ψ(M1) + ψ(M2), (1.3)

homogeneous if

ψ(λM) = |λ|ψ(M), λ ∈ R, (1.4)

set additive (or have the maximum property) if

ψ(M1 ∪M2) = max(ψ(M1), ψ(M2)). (1.5)

In what follows, let X be a Banach space, C a nonempty closed convex subset

of X, F : C → 2C a multivalued mapping and Ω a nonempty subset of C. For any

M ⊆ C we set

F (1,Ω)(M) = F (M), F (n,Ω)(M) = F(

co(

F (n−1,Ω)(M) ∪ Ω))

, (1.6)

for n = 2, 3, . . . .

Definition 1.3. LetX be a Banach space and ψ be a measure of weak noncompact-

ness on X. Let C be a nonempty closed convex subset of X and Ω be a nonempty

weakly compact subset of C. Let F : C → 2C be a bounded multivalued mapping

(that is it takes bounded sets into bounded ones). We say that F is a ψ-convex-power

condensing operator of order n0 with support Ω if for any bounded setM ⊆ C with

ψ(M) > 0 we have

ψ(F (n0,Ω)(M)) < ψ(M). (1.7)

Obviously, F : C → 2C is ψ-condensing if and only if it is ψ- convex-power condens-

ing operator of order one with support Ω.

Remark 1.2.

(1) Let x0 be in C. We abbreviate F (n,x0) to F (n,x0).

(2) If we take Ω = x0 with x0 ∈ C in Definition 1.3 we obtain a multivalued

version of [20] (see also [24]) for the weak topology.

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2. Fixed Point Theorems

We begin this section by presenting a very general Schauder type fixed point theorem

for noncompact Ukc -admissible maps. Before to state our result, recall that the weak

topology on a topological vector space E is the weakest topology (the topology with

the fewest open sets) such that all elements of E′ (the topological dual of E) remain

continuous. Explicitly, a subbase for the weak topology is the collection of sets of

the form f−1(U) where f ∈ E′ and U is an open subset of the base field. A mapping

F : D(F ) ⊆ E → E is said to be weakly compact if F (D(F )) is relatively weakly

compact.

Theorem 2.1. Let C be a nonempty closed convex subset of a locally convex linear

Hausdorff topological space E and F ∈ Uκc (C,C) a weakly compact map. In addition

assume the Krein-Smulian property is satisfied and suppose the following holds:

F (D) is relatively compact for any weakly compact subset D of C. (2.1)

Then F has a fixed point.

Remark 2.1. The Krein-Smulian property states that the closed convex hull of a

weakly compact set is weakly compact.

Remark 2.2. If E is a Banach space then we know [6] that the Krein-Smulian

property holds. For other examples see [7] and [9] .

Proof. Let D = co(F (C)). The Krein-Smulian property guarantees that D is

weakly compact. Also notice D ⊆ C. This implies F (D) ⊆ F (C) ⊆ D. Finally

note F ∈ Uκc (D,D) is a compact map since (2.1) holds. Now apply Theorem 1.1 to

get a fixed point.

As an easy consequence of Theorem 2.1 we recapture the following result which

was proved in [13].

Corollary 2.1. Let C be a nonempty closed convex subset of a Banach space E

and F : C → C a continuous weakly compact map. In addition suppose the following

holds:

if (xn)n∈N is a weakly convergent sequence in C then (Fxn)n∈N

has a strongly convergent subsequence in C

(2.2)

Then F has a fixed point.

Proof. The result follows from Theorem 2.1 once we show (2.1) holds. Let D ⊆ C

be weakly compact and let us look at F (D). Consider a sequence (yn)n∈N in F (D).

For each n ∈ N there exists xn ∈ D with yn = Fxn. Now the Eberlein-Smulian

theorem [7] guarantees that there exists a subsequence S of N so that (xn)n∈S is

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a weakly convergent sequence. Now (2.2) guarantees that (Fxn)n∈S has a strongly

convergent subsequence. Thus F (D) is relatively compact.

Corollary 2.2. Let E be a Banach space and ψ a regular set additive measure of

weak noncompactness on E. Let C be a nonempty closed convex subset of E and Ω

a nonempty weakly compact subset of C. Suppose F ∈ Uκc (C,C) is ψ-convex power

condensing of order n0 with support Ω, F (C) is bounded and (2.1) holds. Then F

has a fixed point.

Proof. Let

F = A ⊆ C, co(A) = A, Ω ⊆ A and F (A) ⊆ A.

The set F is nonempty since C ∈ F . Set

M =⋂

A∈F

A

Now we show that for any positive integer n we have

P(n) M = co(

F (n,Ω)(M) ∪ Ω)

.

To see this, we proceed by induction. ClearlyM is a closed convex subset of C which

contains Ω and F (M) ⊆M. Thus M ∈ F . This implies co(F (M) ∪Ω) ⊆M. Hence

F (co(F (M) ∪ Ω)) ⊆ F (M) ⊆ co(F (M) ∪ Ω). Consequently co(F (M) ∪ Ω) ∈ F .

Hence M ⊆ co(F (M) ∪ Ω). As a result co(F (M) ∪ Ω) = M. This shows that

P(1) holds. Let n be fixed and suppose P(n) holds. This implies F (n+1,Ω)(M) =

F (co(

F (n,Ω)(M) ∪ Ω)

= F (M). Consequently,

co(

F (n+1,Ω)(M) ∪Ω)

= co(F (M) ∪ Ω) =M. (2.3)

As a result

co(

F (n0,Ω)(M) ∪ Ω)

=M. (2.4)

Using the properties of the measure of weak noncompactness we get

ψ(M) = ψ(co(

F (n0,Ω)(M) ∪Ω)

) = ψ(F (n0,Ω)(M)),

which yields that M is weakly compact. Thus F ∈ Uκc (M,M) is weakly compact.

Now apply Theorem 2.1 to get a fixed point.

Our next result is a nonlinear alternative of Leray-Schauder type for Ukc - admis-

sible maps that satisfy a condition of (2.1) type.

Theorem 2.2. Let E be a locally convex linear Hausdororff topological space with

the Krein-Smulian property holding. Let C be a closed convex subset of E, U a

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convex subset of C and U an open (strong topology) subset of E with 0 ∈ U. Suppose

F ∈ Ukc (U,C) is weakly compact and

F (D) is relatively compact for any weakly compact subset D of U (2.5)

holds. Also assume

x 6= λFx for x ∈ ∂CU and λ ∈ (0, 1); (2.6)

here ∂CU denotes the boundary (strong topology) of U in C. Then F has a fixed

point.

Proof. First note intCU = U (interior in the strong topology) since U is open in

C so as a result ∂CU = ∂EU ; here ∂EU denotes the boundary of U in E. Let µ be

the Minkowski functional on U and let r : E → U be given by

r(x) =x

max1, µ(x), for x ∈ E. (2.7)

Note r : E → U is continuous. Also rF ∈ Ukc (U,U) since Uk

c is closed under com-

position. On the other hand, it is easily seen that

rF (U ) ⊆ co(

F (U) ∪ 0)

. (2.8)

Since F (U) is relatively weakly compact then, by the Krein-Smulian property, we

infer that co(

F (U) ∪ 0)

is weakly compact. In view of (2.8) we deduce that rF is

weakly compact. Now we show that rF satisfies (2.1). To see this, let D be a weakly

compact subset of U. From our assumption we have F (D) is relatively compact and

so rF (D) is. By Theorem 2.1 there exists x ∈ U with x = rF (x). Thus x = r(y)

with y = F (x) and x ∈ U = U ∪ ∂U (note intCU = U since U is also open in

C). Now either y ∈ U or y /∈ U. If y ∈ U then r(y) = y so x = y = F (x), and

we are finished. If y /∈ U then r(y) = yµ(y) with µ(y) > 1. Then x = λy with

0 < λ = 1µ(y) < 1; note x ∈ ∂CU since µ(x) = µ(λy) = 1. This contradicts (2.6).

We can improve Theorem 2.2 if E is a Banach space.

Theorem 2.3. Let E be a Banach space and ψ be a regular subadditive measure of

weak noncompactness on E. Let C be a closed convex subset of E, U a convex subset

of C and U an open (strong topology) subset of E with 0 ∈ U. Suppose F : C → 2C

is ψ-convex-power condensing of order n0 ∈ N with support 0, F ∈ Ukc (U,C),

F (U) is bounded and (2.5) holds. Also assume

x 6= λFx for x ∈ ∂CU and λ ∈ (0, 1); (2.9)

here ∂CU denotes the boundary (strong topology) of U in C. Then F has a fixed

point.

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Proof. Let µ be the Minkowski functional on U and let r : E → U be given by

r(x) =x

max1, µ(x), for x ∈ E. (2.10)

Note r : E → U is continuous. Also rF ∈ Ukc (U,U) since U

kc is closed under compo-

sition. Now we show that rF is is ψ-convex-power condensing of order n0 ∈ N with

support 0. To see this, notice first

rF (M) ⊆ co (F (M) ∪ 0) , (2.11)

for any bounded M ⊆ U. Hence

(rF )(1,0)(M) = rF (M) = rF (1,0)(M) ⊆ co(

F (1,0)(M) ∪ 0)

, (2.12)

Consequently

(rF )(2,0)(M) = rF(

co(

(rF )(1,0)(M) ∪ 0))

⊆ rF(

co(

F (1,0)(M) ∪ 0))

= rF (2,0)(M),

by induction,

(rF )(n0,0)(M) = rF(

co(

(rF )(n0−1,0)(M) ∪ 0))

⊆ rF(

co(

rF (n0−1,0)(M) ∪ 0))

⊆ rF(

co(

F (n0−1,0)(M) ∪ 0))

= rF (n0,0)(M).

Consequently

(rF )(n0,0)(M) ⊆ co(

F (n0,0)(M) ∪ 0)

. (2.13)

Thus for any bounded subset M of U with ψ(M) > 0 we have

ψ((rF )(n0,0)(M)) ≤ ψ(co(

F (n0,0)(M) ∪ 0)

) = ψ(F (n0,0)(M)) < ψ(M).

This proves that rF is ψ-power-convex condensing of order n0 with support 0. On

the other hand, rF verifies (2.5) since F does and r is continuous. Also, the reasoning

in the proof of Theorem 2.2 yields that rF is weakly compact. By Theorem 2.1 there

exists x ∈ U with x = rF (x). Thus x = r(y) with y = F (x) and x ∈ U = U ∪ ∂U.

Now either y ∈ U or y /∈ U. If y ∈ U then r(y) = y so x = y = F (x), and

we are finished. If y /∈ U then r(y) = yµ(y) with µ(y) > 1. Then x = λy with

0 < λ = 1µ(y) < 1; note x ∈ ∂CU since µ(x) = µ(λy) = 1. This contradicts (2.9).

Theorem 2.4. Let E be a complete locally convex linear Hausdorff topological

space. Let C be a closed convex subset of E, U an open subset of C with 0 ∈ U

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and F : U → K(C) a weakly compact AcAp map. In addition assume the Krein-

Smulian property is satisfied and suppose that (2.5) holds. Then either

F has a fixed point, (2.14)

or

there is a point u ∈ ∂CU and λ ∈ (0, 1) with u ∈ λFu; (2.15)

here ∂CU is the boundary of U in C.

Proof. Suppose (2.15) does not occur and F does not have a fixed point on ∂CU

(otherwise we are finished since (2.14) occurs). Let

M = x ∈ U : x ∈ λFx for some λ ∈ [0, 1].

The setM is nonempty since 0 ∈ U. AlsoM is closed. Indeed let (xα) be a net ofM

which converges to some x ∈ U and let (λα) be a net of [0, 1] satisfying xα ∈ λnFxα.

Then for each α there is a zα ∈ Fxα with xα = λαzα. By passing to a subnet if

necessary, we may assume that (λα) converges to some λ ∈ [0, 1] and λα 6= 0 for all

α. This implies that the net (zα) converges to some z ∈ U with x = λz. Since F is

closed then z ∈ F (x). Hence x ∈ λFx and therefore x ∈ M. Thus M is closed. We

now claim that M is weakly compact. Since

M ⊆ co(F (M) ∪ 0) (2.16)

and F (M) is weakly compact, then the Krein-Smulian property guarantees that M

is weakly compact. This proves our claim. Now (2.5) implies that F (M) is relatively

compact. Now (2.16) guarantees that M is compact (recall E is complete). From

our assumptions we have M ∩ ∂CU = ∅. Since C as a subset of a locally convex

Hausdorff topological linear space is completely regular [8] there is a continuous

mapping ρ : U → [0, 1] with ρ(M) = 1 and ρ(∂CU) = 0. Let

J(x) =

ρ(x)F (x), x ∈ U,

0, x ∈ C \ U.

Clearly J : C → K(C) is closed since F is closed and ρ is continuous. Moreover, for

any D ⊆ C we have

J(D) ⊆ co(F (D ∩ U) ∪ 0). (2.17)

Taking into account that F is weakly compact and using (2.17) and the Krein-

Smulian property we infer that J is weakly compact. Now we show that J verifies

(2.1). To see this let D ⊆ C be weakly compact. In view of our assumptions we

have F (D∩U ) is relatively compact. Now (2.17) guarantees that J(D) is relatively

compact. This proves that J verifies (2.1). If F is approximable then from [17] we

know that J is approximable. If F is acyclic then it is clear that J is acyclic. In

both cases J : C → K(C) is an AcAp map. Invoking Theorem 2.1 there exists x ∈ C

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such that x ∈ J(x). Now x ∈ U since 0 ∈ U. Consequently x ∈ ρ(x)F (x) and so

x ∈M. This implies ρ(x) = 1 and so x ∈ F (x).

Our next result is a Furi-Pera type fixed point theorem for weakly compact

Ukc -admissible maps.

Theorem 2.5. Let E be a complete metrizable locally convex topological vector

space, Q a closed convex subset of E and 0 ∈ Q. Suppose F ∈ Ukc (Q,E) is a weakly

compact map and assume the following conditions hold:

there exists a continuous retraction r : E → Q with r(z) ∈ ∂Q for z ∈ E\Q

and r(D) is relatively weakly compact for any weakly compact subset D of E.

(2.18)

F (D) is relatively compact for any weakly compact subset D of Q. (2.19)

if (xj , λj)+∞j=1 is a sequence of ∂Q× [0, 1] converging to (x, λ)with x ∈ λFx

and 0 ≤ λ < 1, then λjFxj ⊆ Q for j sufficiently large.

(2.20)

Then F has a fixed point in Q.

Proof. Let r be as described in (2.18). Clearly Fr ∈ Ukc (E,E). Consider

B = x ∈ E : x ∈ Fr(x)

Firstly, B 6= ∅. To see this, notice that Fr is a weakly compact map since r(E) ⊆ Q

and F is weakly compact. Now let D be a weakly compact subset of E. Notice

that r(D) is relatively weakly compact, since (2.18) holds. From hypothesis (2.19)

it follows that Fr(D) is relatively compact. Now Theorem 2.1 guarantees that Fr

has a fixed point, and therefore B 6= ∅. In addition, since F is closed, we have that

B is closed. Also notice that

B ⊆ Fr(B) ⊆ F (Q). (2.21)

This implies that B relatively weakly compact. Now (2.18) guarantees that r(B) is

relatively weakly compact. This together with (2.19) implies that Fr(B) is relatively

compact. Now from (2.21) it follows that B is compact. It remains to show that

B ∩ Q 6= ∅. To do this we argue by contradiction. Suppose that B ∩Q = ∅. Then,

since B is compact and Q is closed, there exists a δ > 0 with dist(B,Q) > δ. Choose

m ∈ 1, 2, . . . with 1 < δm. Define

Ui = x ∈ E : d(x,Q) <1

i for i ∈ m,m+ 1, . . .;

here d is the metric associated with E. Fix i ∈ m,m+1, . . .. Since dist(B,Q) > δ,

we see that B ∩ Ui = ∅. In addition, Ui is open and convex, 0 ∈ Ui and Fr ∈

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Ukc (Ui, E) is a weakly compact map and verifies (2.5). Theorem 2.2 guarantees,

since B ∩ Ui = ∅, that there exists

(yi, λi) ∈ ∂Ui × (0, 1) with yi ∈ λiFr(yi).

We can do this for each i ∈ m,m+ 1, . . .. Thus we have

λiFr(yi) * Q for each i ∈ m,m+ 1, . . .. (2.22)

We now look at

H = x ∈ E : x ∈ λFr(x) for some λ ∈ [0, 1]. (2.23)

Notice that H 6= ∅ is closed and weakly compact, since F ∈ Ukc (Q,E) is a weakly

compact map. Also, taking into account the fact that

H ⊆ co (Fr(H) ∪ 0) , (2.24)

the use of the Krein-Smulian property gives that H is relatively weakly compact.

From (2.18) it follows that r(H) is relatively weakly compact. The use of (2.19)

implies that Fr(H) is relatively compact. Now (2.24) guarantees that H is compact.

This, together with

d(yj , Q) =1

jand |λj | ≤ 1 for j ∈ m,m+ 1, . . .,

implies that we may assume without loss of generality that

λj → λ∗ ∈ [0, 1] and yj → y∗ ∈ ∂Q.

In addition we have yj ∈ λjFr(yj) with F closed, and so y∗ ∈ λ∗Fr(y∗). Note that

λ∗ 6= 1 since B ∩Q = ∅. Hence 0 ≤ λ∗ < 1. However, (2.20), with

xj = r(yj) ∈ ∂Q and x = y∗ = r(y∗),

implies that λjFr(yj) ⊆ Q for j sufficiently large. This contradicts (2.22).

Thus B ∩Q 6= ∅, so there exists x ∈ Q with x ∈ Fr(x) = F (x).

Remark 2.3. If 0 ∈ int(Q) then we can choose r : E → Q in the statement of

Theorem 2.5 as

r(x) =x

max1, µ(x), for x ∈ E,

where µ in the Minkowski functional on Q. Clearly r(z) ∈ ∂Q for z ∈ E\Q. Now

consider a weakly compact subset D of E. Note

r(D) ⊆ co(D ∪ 0). (2.25)

The Krein-Smulian property guarantees that r(D) is relatively weakly compact.

We can improve Theorem 2.5 if E is a Banach space.

Theorem 2.6. Let E be a Banach space and ψ a regular set additive measure of

weak noncompactness on E. Let Q be a closed convex subset of E with 0 ∈ Q.

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Suppose F : E → 2E is ψ-power-convex condensing of order n0 ∈ N with support

0 and assume F ∈ Ukc (Q,E), F (Q) is bounded and (2.19) and (2.20) hold. Also

suppose the following condition holds:

there exists a continuous retraction r : E → Q with r(z) ∈ ∂Q for z ∈ E\Q

and r(D) ⊆ co(D ∪ 0) for any bounded subset D of E.

(2.26)

Then F has a fixed point in Q.

Proof. Let r be as described in (2.26). Consider

B = x ∈ E : x ∈ Fr(x)

Firstly, B 6= ∅. To see this, let D be a weakly compact subset of E. Using (2.26)

together with the Krein-Smulian property we infer that r(D) is relatively weakly

compact. Now (2.19) guarantees that Fr(D) is relatively compact. This proves that

Fr maps weakly compact sets into relatively compact sets. Now we show that Fr

is ψ-power-convex condensing of order n0 ∈ N with support 0. To see this, let D

be a bounded subset of E and set D′ = co(D ∪ 0). Then, keeping in mind (2.26)

we obtain

(Fr)(1,0)(D) ⊆ F (D′),

(Fr)(2,0)(D) = Fr(

co(

(Fr)(1,0)(D) ∪ 0))

⊆ Fr (co (F (D′) ∪ 0))

⊆ F (co (F (D′) ∪ 0))

= F (2,0)(D′),

and by induction,

(Fr)(n0,0)(D) = Fr(

co(

(Fr)(n0−1,0)(D) ∪ 0))

⊆ Fr(

co(

F (n0−1,0)(D′) ∪ 0))

⊆ F(

co(

F (n0−1,0)(D′) ∪ 0))

= F (n0,0)(D′).

Thus

ψ(

(Fr)(n0,0)(D))

≤ ψ(

F (n0,0)(D′))

< ψ(D′) = ψ(D),

whenever ψ(D) 6= 0. Now Corollary 2.2 guarantees that Fr has a fixed point, and

therefore B 6= ∅. In addition, since F is closed, we have that B is closed. Also notice

that

B ⊆ Fr(B) (2.27)

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Thus

B ⊆ Fr(B) ⊆ Fr(co (Fr(B) ∪ 0) = (Fr)(2,0)(B),

and by induction

B ⊆ Fr(B) ⊆ Fr(co(

(Fr)(n0−1,0)(B) ∪ 0)

= (Fr)(n0,0)(B).

Consequently

ψ(B) ≤ ψ(

(Fr)(n0,0)(B))

.

Since Fr is ψ-power-convex condensing of order n0 ∈ N with support 0 then

ψ(B) = 0 and so B is relatively weakly compact. The use of (2.26) together with

the Krein-Smulian property gives that r(B) is relatively weakly compact. Therefore

Fr(B) is relatively compact. Now from (2.27) it follows that B is compact. It

remains to show that B ∩ Q 6= ∅. To do this we argue by contradiction. Suppose

that B∩Q = ∅. Then, since B is compact and Q is closed, there exists a δ > 0 with

dist(B,Q) > δ. Choose m ∈ 1, 2, . . . with 1 < δm and let Ui be as in Theorem

2.5. Essentially the same reasoning as in Theorem 2.5 guarantees that

λiFr(yi) * Q for each i ∈ m,m+ 1, . . .. (2.28)

Let H be as in Theorem 2.5. Notice that H 6= ∅ is closed since F is closed. Also,

taking into account the fact that

H ⊆ co (Fr(H) ∪ 0) , (2.29)

we deduce that

Fr(H) ⊆ Fr (co (Fr(H) ∪ 0)) = (Fr)(2,0)(H). (2.30)

Combining (2.29) and (2.30) we arrive at

H ⊆ co(

(Fr)(2,0)(H) ∪ 0)

.

By induction

Fr(H) ⊆ Fr(

co(

(Fr)(n0−1,0)(H) ∪ 0))

= (Fr)(n0,0)(H).

Thus

H ⊆ co(

(Fr)(n0,0)(H) ∪ 0)

.

This implies that

ψ(H) ≤ ψ(

co(

(Fr)(n0,0)(H) ∪ 0))

= ψ(

(Fr)(n0,0)(H))

.

Since Fr is ψ-power-convex condensing of order n0 ∈ N with support 0 then

ψ(H) = 0 and so H is relatively weakly compact. Therefore Fr(H) is relatively

compact. Now (2.29) guarantees that H is compact. As in Theorem 2.5 we may

assume that

λj → λ∗ ∈ [0, 1] and yj → y∗ ∈ ∂Q.

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This implies that y∗ ∈ λ∗Fr(y∗) with 0 ≤ λ∗ < 1. This together with (2.20)

implies that λjFr(yj) ⊆ Q for j sufficiently large. This contradicts (2.28). Thus

B ∩Q 6= ∅, so there exists x ∈ Q with x ∈ Fr(x) = F (x).

Remark 2.4. If 0 ∈ int(Q) then we can choose r : E → Q in the statement of

Theorem 2.6 as

r(x) =x

max1, µ(x), for x ∈ E,

where µ in the Minkowski functional on Q. Clearly r(z) ∈ ∂Q for z ∈ E\Q and

r(D) ⊆ co(D ∪ 0) for any bounded subset D of E.

References

1. R. P. Agarwal, D. O’Regan and S. Park, Fixed point theory for multimaps in extensiontype spaces, J. Korean Math. Soc. 39 (2002) 579–591.

2. R. P. Agarwal, D. O’Regan, Essential Ukc - type maps and Birkhoff-Kellogg theorems

J. of Appl. Math. Stoch. Anal. (2004) 1–8.3. O. Arino, S. Gautier and J. P. Penot, A fixed point theorem for sequentially continuous

mappings with applications to ordinary differential equations, Funkc. Ekvac. 27 (1984)273–279.

4. J. Banas, J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl.

151 (1988) 213–224.5. H. Ben-El-Mechaiekh, P. Deguire, Approachability and fixed points for nonconvex set

valued maps, J. Math. Anal. Appl. 170 (1992) 477–500.6. N. Dunford, J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience

Publishers, New York, 1958.7. R. E. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and

Winston, 1965.8. R. Engelking, General Topology, Heldermann Verlag, Berlin (1989).9. K. Floret, Weakly compact sets, Lecture Notes in Mathematics, 801, Springer-Verlag,

Berlin (1980).10. J. Garcia-Falset, Existence of fixed points and measure of weak noncompactness, Non-

linear Anal. 71 (2009) 2625–2633.11. R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981) 81–86.12. V. Klee, Leray-Schauder theory without local convexity, Math. Ann. 141 (1960) 286–

296, Corrections, Math. Ann. 145 (1962), 464–465.13. K. Latrach, M. A. Taoudi and A. Zeghal, Some fixed point theorems of the Schauder

and Krasnosel’skii type and application to nonlinear transport equations, J. Differ-

ential Equations 221 no. 1 (2006) 256–271.14. K. Latrach, M. A. Taoudi, Existence results for a generalized nonlinear Hammerstein

equation on L1-spaces, Nonlinear Anal. 66 (2007) 2325–2333.

15. M. Nagumo, Degree of mappings in convex linear topological spaces, Amer. J. Math.

73 (1951) 497–511.16. D. O’Regan, Fixed point theory on extension type spaces and essential maps on topo-

logical spaces, Fixed Point Theory Appl. (2004) 13–20.17. D. O’Regan, Fixed point theory for multivalued mappings in completely regular topo-

logical vector spaces, Math. Comput. Modelling, Vol. 28, No. 10 (1998) 7–15.18. D. O’Regan, Furi-Pera type theorems for U

kc -admissible maps of Park, Mathematical

Proceeding of the Royal Irish Academy, 102 A (2) (2002) 163–173.

Asi

an-E

urop

ean

J. M

ath.

201

1.04

:373

-387

. Dow

nloa

ded

from

ww

w.w

orld

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MO

NA

SH U

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ER

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Y o

n 10

/26/

12. F

or p

erso

nal u

se o

nly.



19. S. Park, A unified fixed point theory of multimaps on topological vector spaces, J.Korean Math. Soc. 35 (1998) 803–829.

20. J. Sun and X, Zhang, The fixed point theorem of convex-power condensing operatorand applications to abstract semilinear evolution equations, Acta Math. Sinica 48

(2005) 339–446.21. M. A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an

unbounded interval, Nonlinear Anal. 71 (2009) 4131–4136.22. M. A. Taoudi, N. Salhi and B. Ghribi, Integrable solutions of a mixed type operator

equation, Appl. Math. Comput. 216 (2010) 1150–1157.23. E. Zeidler, Nonlinear Functional Analysis and its Applications, I : Fixed Point Theo-

rems, Springer-Verlag, New York, 1986.24. Guowei Zhang, Tongshan Zhang, Tie Zhang, Fixed point theorems of Rothe and

Altman types about convex-power condensing operator and application, Appl. Math.

Comput. 214 (2009) 618–623.

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/26/

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FIXED POINT THEOREMS FOR GENERAL CLASSES OF MAPS ACTING ON TOPOLOGICAL VECTOR SPACES

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